On the convergence of collocation methods for boundary integral equations on polygons
HTML articles powered by AMS MathViewer
- by Martin Costabel and Ernst P. Stephan PDF
- Math. Comp. 49 (1987), 461-478 Request permission
Abstract:
The integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions. Convergence proofs and error estimates for these approximation methods have been only available in the following cases: Fredholm integral equations of the second kind [4], [7], one-dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves [2], [3], [17], [26]—[29], and some special results on the classical Neumann integral equation of potential theory for polygonal plane domains [5], [8], [9]. Here we give convergence proofs for collocation with piecewise linear trial functions for Neumann’s integral equation and Symm’s integral equation on plane curves with corners. We derive asymptotic error estimates in Sobolev norms and analyze the effect of graded meshes.References
- Philip M. Anselone, Collectively compact operator approximation theory and applications to integral equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. With an appendix by Joel Davis. MR 0443383
- Douglas N. Arnold and Wolfgang L. Wendland, On the asymptotic convergence of collocation methods, Math. Comp. 41 (1983), no. 164, 349–381. MR 717691, DOI 10.1090/S0025-5718-1983-0717691-6
- Douglas N. Arnold and Wolfgang L. Wendland, The convergence of spline collocation for strongly elliptic equations on curves, Numer. Math. 47 (1985), no. 3, 317–341. MR 808553, DOI 10.1007/BF01389582
- Kendall E. Atkinson, A survey of numerical methods for the solution of Fredholm integral equations of the second kind, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. MR 0483585 K. E. Atkinson & F. R. de Hoog, "Collocation methods for a boundary integral equation on a wedge," in Treatment of Integral Equations by Numerical Methods (C. T. H. Baker and B. F. Miller, eds.), Academic Press, New York, 1983.
- Ivo Babuška and Michael B. Rosenzweig, A finite element scheme for domains with corners, Numer. Math. 20 (1972/73), 1–21. MR 323129, DOI 10.1007/BF01436639
- Christopher T. H. Baker, The numerical treatment of integral equations, Monographs on Numerical Analysis, Clarendon Press, Oxford, 1977. MR 0467215 G. Bruhn & W. L. Wendland, "Über die näherungsweise Lösung von linearen Funktionalgleichungen," in Funktionalanalysis, Approximationstheorie, Numerische Mathematik (L. Collatz and H. Ehrmann, eds.), Birkhäuser, Basel, 1967. G. A. Chandler & I. G. Graham, "Product integration-collocation methods for non-compact integral operator equations." (To appear.)
- Martin Costabel, Boundary integral operators on curved polygons, Ann. Mat. Pura Appl. (4) 133 (1983), 305–326. MR 725031, DOI 10.1007/BF01766023
- Martin Costabel and Ernst Stephan, Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, Mathematical models and methods in mechanics, Banach Center Publ., vol. 15, PWN, Warsaw, 1985, pp. 175–251. MR 874845 M. Costabel & E. P. Stephan, "The method of Mellin transformation for boundary integral equations on curves with corners," Numerical Solution of Singular Integral Equations, IMACS, 1984, pp. 95-100.
- Martin Costabel and Ernst Stephan, A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl. 106 (1985), no. 2, 367–413. MR 782799, DOI 10.1016/0022-247X(85)90118-0
- Martin Costabel, Ernst Stephan, and Wolfgang L. Wendland, On boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 2, 197–241. MR 728434
- J. Elschner, Galerkin methods with splines for singular integral equations over $(0,\,1)$, Numer. Math. 43 (1984), no. 2, 265–281. MR 736084, DOI 10.1007/BF01390127
- I. C. Gohberg and I. A. Fel′dman, Convolution equations and projection methods for their solution, Translations of Mathematical Monographs, Vol. 41, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by F. M. Goldware. MR 0355675
- Roland Hagen and Bernd Silbermann, A finite element collocation method for bisingular integral equations, Applicable Anal. 19 (1985), no. 2-3, 117–135. MR 800163, DOI 10.1080/00036818508839538
- Stefan Hildebrandt and Ernst Wienholtz, Constructive proofs of representation theorems in separable Hilbert space, Comm. Pure Appl. Math. 17 (1964), 369–373. MR 166608, DOI 10.1002/cpa.3160170309 M. A. Krasnoselskiǐ, et al., Approximate Solution of Operator Equations, Noordhoff, Groningen, 1972.
- J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350177 V. G. Maz’ya & T. O. Shaposhnikova, "Change of variables as an operator in a pair of Sobolev spaces," Vestnik Leningrad Univ. Math., v. 15, 1983, pp. 53-58.
- Siegfried Prössdorf, Ein Lokalisierungsprinzip in der Theorie der Spline-Approximationen und einige Anwendungen, Math. Nachr. 119 (1984), 239–255 (German). MR 774194, DOI 10.1002/mana.19841190123
- Siegfried Prössdorf and Bernd Silbermann, Projektionsverfahren und die näherungsweise Lösung singulärer Gleichungen, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1977 (German). Mit einer englischen und einer russischen Zusammenfassung; Teubner-Texte zur Mathematik. MR 0494817
- Siegfried Prössdorf and Bernd Silbermann, Gestörte Projektionsverfahren und einige ihrer Anwendungen, Theory of nonlinear operators (Proc. Fifth Internat. Summer School, Central Inst. Math. Mech. Acad. Sci. GDR, Berlin, 1977) Abh. Akad. Wiss. DDR, Abt. Math. Naturwiss. Tech., 1978, vol. 6, Akademie-Verlag, Berlin, 1978, pp. 229–237 (German, with English summary). MR 540463 S. Prössdorf & J. Elschner, "Finite element methods for singular integral equations on an interval," Engrg. Anal., v. 1, 1984, pp. 83-87.
- Siegfried Prössdorf and Andreas Rathsfeld, A spline collocation method for singular integral equations with piecewise continuous coefficients, Integral Equations Operator Theory 7 (1984), no. 4, 536–560. MR 757987, DOI 10.1007/BF01238865
- J. Saranen and W. L. Wendland, On the asymptotic convergence of collocation methods with spline functions of even degree, Math. Comp. 45 (1985), no. 171, 91–108. MR 790646, DOI 10.1090/S0025-5718-1985-0790646-3
- G. Schmidt, On spline collocation for singular integral equations, Math. Nachr. 111 (1983), 177–196. MR 725777, DOI 10.1002/mana.19831110108
- W. L. Wendland, Boundary element methods and their asymptotic convergence, Theoretical acoustics and numerical techniques, CISM Courses and Lect., vol. 277, Springer, Vienna, 1983, pp. 135–216. MR 762829 W. L. Wendland, "On the spline approximation of singular integral equations and one-dimensional pseudodifferential equations on closed curves," Numerical Solution of Singular Integral Equations, IMACS, 1984, pp. 113-119.
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp. 49 (1987), 461-478
- MSC: Primary 65R20; Secondary 65D07, 65N35
- DOI: https://doi.org/10.1090/S0025-5718-1987-0906182-9
- MathSciNet review: 906182